Professor: Professor R. J. Williams
Time: M 4.40--6.10 p.m., W 5.10--6.10 p.m.
Place: AP&M 7421
Office Hours: M & W, 3--4 pm, AP&M 6121.

DESCRIPTION: This course is an introduction to the mathematics of financial models. The aim is to provide students with an introduction to some basic models of finance and the associated mathematical machinery.

OUTLINE: The course will begin with the development of the basic ideas of hedging and pricing by arbitrage in the discrete time setting of binomial tree models. Key probabilistic concepts of conditional expectation, martingale, change of measure, and representation, will all be introduced first in this simple framework as a bridge to the continuous model setting. Mathematical fundamentals for the development and analysis of continous time models will be covered, including Brownian motion, stochastic calculus, change of measure, martingale representation theorem. These will then be combined to develop the Black-Scholes option pricing formula. Pricing and hedging for European and American call options will be discussed. Attention will then turn to models of the interest rate market. Various models will be discussed, including the Heath-Jarrow-Morton and Cox-Ingersoll-Ross models. As time permits, more general models and extensions will be described.

COMPUTER MODULES: These will complement the theoretical material presented in the course. Students may wish to enrol in Math 161 in the Fall of 1998, which will include an Introduction to Mathematica.


  • Mathematics of Financial Markets, R. J. Elliott and P. E. Kopp, Springer, 1998. This book is now available in the bookstore.

    Background in economics/finance:

  • Investment Science, David G. Luenberger, Oxford University Press, 1998.
  • Financial Economics, H. H. Panjer (ed.), The Actuarial Foundation, Schaumburg, Illinois, 1998.
  • Options, Futures and other Derivative Securities, J. Hull, Prentice Hall, 1993.
    Mathematics of Finance: Stochastic Approaches
  • Financial calculus, Martin Baxter and Andrew Rennie, Cambridge University Press, 1996.
  • Introduction to Stochastic Calculus Applied to Finance, D. Lamberton and B. Lapeyre, Chapman and Hall, 1996.
  • An Introduction to the Mathematics of Financial Derivatives, Salih N. Neftci, Academic Press, 1996.
  • Steven Shreve's Lectures on Stochastic Calculus and Finance, Prepared by P. Chalasani and S. Jha.
    Mathematics of Finance: PDE Approach
  • The Mathematics of Financial Derivatives: A student introduction, Paul Wilmott, et al., Cambridge University Press, 1995.
    Mathematics of Finance: more advanced stochastic theory
  • Martingale methods in financial modelling, M. Musiela and M. Rutkowski, Springer, 1998.
  • Methods of mathematical finance, I. Karatzas and S. Shreve, Springer, 1998.


  • Mathematical Finance talk by R. J. Elliott at UCSD
  • Probability Seminar talk by Bernt Oksendal: "A short introduction to Malliavin calculus and its application to finance", Thursday, February 25, 1999, 10.10am, AP&M 6218. For Oksendal's notes on Malliavin calculus, click here.
  • Professor Harry Markowitz, Leverage and Growth: with implications on hedgefunds.
    Thursday, March 11, 1999, Price Center Theater, 5 p.m.

    LINKS TO RELATED WEB SITES (under construction):

    Please direct any questions to Professor Ruth J. Williams, email: